Termination Competition The Termination Competition Claude March´ e Hans Zantema Orsay, France Eindhoven, The Netherlands June 26th, 2007
Termination Competition Short history • WST’03, Valencia: initiated by Albert Rubio • Termination Problems Data Base • Competition/Exhibition of termination tools • Since 2004: ‘automatic’ competition • tools run fully automatically • results available ‘live’ on a web page • Goals of such a competition: • stimulate research on termination techniques • put emphasis on automation • provide a standard to compare termination techniques
Termination Competition The TPDB • Problems of the TPDB: 3 syntax • String Rewriting System (SRS) • Term Rewriting System (TRS) • Logic Program (LP) • SRS sub-category: relative termination • TRS sub-categories: • modulo theory (only AC in current TPBD) • reduction strategies: innermost, outermost, context-sensitive • relative termination
Termination Competition The rules • Tools run on all problems of the TPDB they support, on the same computer • Running time is limited (1 minute) • Required output: • “YES”, followed by the text of a termination proof, or • “NO”, followed by the text of a non-termination proof, or • anything else, including time limit reached, is interpreted as “DON’T KNOW” • Score: 1 point for each problem solved
Termination Competition TRS category (not shown: Mu-Term, TTTbox)
Termination Competition SRS category (not shown: CiME, TPA, TTTbox)
Termination Competition A success story • SRS Zantema/z086 : aa → bc bb → ac cc → ab • Unsolved by any tool in 2004 and 2005 • Became RTA open problem #104 • First solved ‘by hand’: [Hofbauer, Waldmann, IPL 06] • Brought up idea of Matrix interpretations • Solved by Jambox tool in 2006 (clear winner of SRS category) • Matrix interpretations on terms [Endrullis, IJCAR 06] • Jambox got second ( 628 ) just behind AProVe ( 638 ) in TRS category
Termination Competition Other evidences of success • Usefulness of back-end SAT solvers for finding solutions to orderings constraints • “AProVe couldn’t remain winner each year without several major improvements”: • applicative TRSs [Giesl et al., FroCos 05] • polynomials with negative coefs [Hirokawa, Middeldorp, AISC 04] • subterm criterion [Hirokawa, Middeldorp, RTA 04] • match-bounds for term rewriting [Geser et al. IC 07] • etc.
Termination Competition Longstanding open problems • Past open problems, now solved, e.g.: • AC-TRSs for integer arithmetic [Contejean et al., RTA 97] for sequent calculus modulo [Deplagne, 00], solved in 2004 by CiME • TRSs for explicit substitutions: [Bonelli] (solved in 2005 by TEPARLA) and TRS/Zantema-z10 (solved by TPA) • Remain open: • Hercules & Hydra battle, only unsolved pb of famous collection “33 problems of termination” [Dershowitz, 1995] • Cohen-Watson [RTA 91] system for arithmetic [RTA LOOP #65]
Termination Competition Other challenges • Emphasized right after the end of the 2006 edition: aaa → bab bbb → aaa • Semantic decreasing argument: • encodings of ‘while loops’ • automatic translations from CS-TRSs, Maude or OBJ programs • SRS challenge: ( SRS/Zantema-z079 ) caa → ac acb → adb ad → daaa bd → bc (essentially rewrites 2 n to 3 n )
Termination Competition Non-termination challenges • No tool able to discover non-looping non-termination • e.g. TRS TRS/HofWald-6 : f ( f ( a , x ) , y ) → f ( f ( x , f ( a , y )) , a ) • or SRS SRS/Zantema-z073 : al → la ra → ar bl → bar rb → lb
Termination Competition Perspectives • Certified proofs of termination • termination tools are complex softwares, hence intrinsically buggy. . . • proof assistants (e.g. Coq, Isabelle) can help for double-checking proofs • Handling programs in ‘real’ languages: • functional: lazy (Haskell), strict (ML) • imperative: C, Java, etc. • Handling numerical computations • built-in integers
Termination Competition The 2007 edition • New category: Haskell programs • New ‘option’: certified proofs of termination What happened? • Some challenges solved! • For details: Johannes Waldmann’s talk on Friday WST workshop 15:15
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